Question

How do you simplify `log_7 9x+log_7x-3log_7 x`?

Answer 1

`log_7(9/x)`

Explanation:

First of all, observe that `3log_7(x)=log_7(x^3)`. So, your expression becomes

`log_7(9x)+log_7(x)-log_7(x^3)`

Now, the sum of two logarithms is the logarithm of the product:

`log_7(a)+log_7(b)=log_7(ab)`

So,

`color(green)(log_7(9x)+log_7(x))-log_7(x^3)=color(green)(log_7(9x^2))-log_7(x^3)`

And the difference of two logarithms is the logarithm of the ratio:

`log_7(a)-log_7(b)=log_7(a/b)`

So,

`log_7(9x^2)-log_7(x^3) = log_7(9x^2/x^3) = log_7(9/x)`